Simplify and expand the following expression: $ \dfrac{4}{q - 8}- \dfrac{3}{5q - 10}+ \dfrac{5}{q^2 - 10q + 16} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the second term: $ \dfrac{3}{5q - 10} = \dfrac{3}{5(q - 2)}$ We can factor the quadratic in the third term: $ \dfrac{5}{q^2 - 10q + 16} = \dfrac{5}{(q - 8)(q - 2)}$ Now we have: $ \dfrac{4}{q - 8}- \dfrac{3}{5(q - 2)}+ \dfrac{5}{(q - 8)(q - 2)} $ The least common multiple of the denominators is: $ (q - 8)(q - 2)$ In order to get the first term over $(q - 8)(q - 2)$ , multiply by $\dfrac{5(q - 2)}{5(q - 2)}$ $ \dfrac{4}{q - 8} \times \dfrac{5(q - 2)}{5(q - 2)} = \dfrac{20(q - 2)}{(q - 8)(q - 2)} $ In order to get the second term over $(q - 8)(q - 2)$ , multiply by $\dfrac{q - 8}{q - 8}$ $ \dfrac{3}{5(q - 2)} \times \dfrac{q - 8}{q - 8} = \dfrac{3(q - 8)}{(q - 8)(q - 2)} $ In order to get the third term over $(q - 8)(q - 2)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{5}{(q - 8)(q - 2)} \times \dfrac{5}{5} = \dfrac{25}{(q - 8)(q - 2)} $ Now we have: $ \dfrac{20(q - 2)}{(q - 8)(q - 2)} - \dfrac{3(q - 8)}{(q - 8)(q - 2)} + \dfrac{25}{(q - 8)(q - 2)} $ $ = \dfrac{ 20(q - 2) - 3(q - 8) + 25} {(q - 8)(q - 2)} $ Expand: $ = \dfrac{20q - 40 - 3q + 24 + 25}{5q^2 - 50q + 80} $ $ = \dfrac{17q + 9}{5q^2 - 50q + 80}$